Question

1. write a two - column proof of corollary 5.1

given: ∠r is a right angle.
prove: ∠s and ∠t are complementary.

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\). For \(\triangle RST\), \(\angle R + \angle S + \angle T = 180^\circ\).

Step2: Substitute \(\angle R\)

Given \(\angle R\) is a right angle, so \(\angle R = 90^\circ\). Substitute into the angle - sum formula: \(90^\circ+\angle S+\angle T = 180^\circ\).

Step3: Solve for \(\angle S+\angle T\)

Subtract \(90^\circ\) from both sides of the equation \(90^\circ+\angle S+\angle T = 180^\circ\). We get \(\angle S+\angle T=180^\circ - 90^\circ=90^\circ\).

Step4: Define Complementary Angles

By definition, two angles are complementary if their sum is \(90^\circ\). Since \(\angle S+\angle T = 90^\circ\), \(\angle S\) and \(\angle T\) are complementary.

Now, we can write the two - column proof:

Statements	Reasons
2. \(\angle R=90^{\circ}\)	Definition of a right angle
3. \(\angle R+\angle S+\angle T = 180^{\circ}\)	Triangle Angle - Sum Theorem
4. \(90^{\circ}+\angle S+\angle T = 180^{\circ}\)	Substitution (Step 2 into Step 3)
5. \(\angle S+\angle T=90^{\circ}\)	Subtraction Property of Equality (Subtract \(90^{\circ}\) from both sides of Step 4)
6. \(\angle S\) and \(\angle T\) are complementary.	Definition of complementary angles (Step 5)

Answer:

The two - column proof is as shown above, proving that \(\angle S\) and \(\angle T\) are complementary.